A178778 Partial sums of walks of length n+1 on a tetrahedron A001998.
1, 3, 7, 17, 42, 112, 308, 882, 2563, 7565, 22449, 66979, 200204, 599514, 1796350, 5385764, 16150725, 48442327, 145307291, 435892341, 1307617966, 3922765316, 11768118792, 35304090646, 105911740487, 317734424289, 953201678533, 2859602644103, 8578803149328
Offset: 0
Examples
a(5) = 1 + 2 + 4 + 10 + 25 + 70 = 112.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (5,-4,-12,21,-9).
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)) )); // G. C. Greubel, Jan 24 2019 -
Mathematica
CoefficientList[Series[(6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)), {x,0,30}], x] (* G. C. Greubel, Jan 24 2019 *) -
PARI
Vec((1-2*x-4*x^2+6*x^3)/((1-x)^2*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 17 2016
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Sage
((6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 24 2019
Formula
a(n) = Sum_{i=0..n} (if i mod 2 = 0 then ((3^((i-2)/2)+1)/2)^2 else 3^((i-3)/2)+(1/4)*(3^(i-2)+1)).
G.f.: (6*x^3-4*x^2-2*x+1) / ((x-1)^2*(3*x-1)*(3*x^2-1)). - Colin Barker, Apr 20 2013
From Colin Barker, May 17 2016: (Start)
a(n) = (-7+3^(1+n)+3^(1/2*(-1+n))*(9-9*(-1)^n+5*sqrt(3)+5*(-1)^n*sqrt(3))+2*(1+n))/8.
a(n) = (2*n + 10*3^(n/2) + 3^(n+1) - 5)/8 for n even.
a(n) = (2*n + 3^(n+1) + 2*3^((n+3)/2) - 5)/8 for n odd.
a(n) = 5*a(n-1) - 4*a(n-2) - 12*a(n-3) + 21*a(n-4) - 9*a(n-5) for n>4.
(End)
Comments